The convergence of IBU has been studied in [1]} and revised in [2]}. In the cases where IBU converges to a MLE, let the limiting estimate of \(\pi _{\mathcal {X}}\)  be called \(\hat{\pi }_{\mathcal {X}}\) , i.e., \(\hat{\pi }_{\mathcal {X}}=\lim \limits _{t\rightarrow \infty }\theta _{t}\) , and as, for an observed set of noisy locations, it is determined completely with the privatizing mechanism \(\mathcal {C}\)  and the starting PMF \(\theta _0\) , let \(\lim \limits _{t\rightarrow \infty }\theta _{t}\)  be functionally denoted as \(\Gamma _{\text{IBU}}(\theta _0,\mathcal {C})\) .
